Published online by Cambridge University Press: 29 May 2025
For a position and a player with a winning move, the pass-value is the largest number of free moves the player can allow the opponent so that after these move(s) the player still has a winning move. For a cell-coloring game such as Hex, the pass-value is equivalently the smallest number of cells the opponent needs to color in order to reach a position where the opponent has a winning move. A move is good if it increases the pass-value. HyperHex is the hypergraph generalization of Hex: each player has a list of winsets, and wins by coloring all cells of any of her winsets. No-draw HyperHex is the maker-breaker restriction of HyperHex: each player's winset list contains every minimal set that intersects all of the other player's winsets (so draws are not possible). For no-draw HyperHex, we consider two good-move proverbs: your opponent's good move is your good move, and it's never too late for a good move.
1. Introduction
At the 2011 Banff International Research Station Workshop on Combinatorial Game Theory, I asked professional 9-dan Ziang Zhujiu (“Jujo”) about the Go proverb your opponent's good move is your good move [?]. His instant response was “it's not always true”, and of course he is right. For example, in Figure 1 each player's winning move (in the opponent's territory) is the opponent's losing move, so no move is good for both a player and their opponent.
This paper is about good moves in no-draw HyperHex, a game that generalizes Hex. We consider the Go proverb above, and the general proverb it's never too late, which when applied to games could be expanded as it's never too late for a good move.
2. No-draw HyperHex
In 1942 Piet Hein invented Polygon, the classic alternate-turn two-player connection game now known as Hex [?]. In 1949 John Nash described the game to David Gale, who built a board that was soon in frequent use in Princeton's Fine Hall [?; ?]. Later John Milnor and independently Claude Shannon [?] and independently Charles Titus (personal communication with Craige Schensted) [?] invented Y, and Shannon created his eponymous (switching) game [?].
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